Obi-wan responds to starwars.wikia.com/wiki/Luke_Skywalker saying R2 belongs to him with, “I don't seem to remember ever owning a droid,” which is technically true, because Obi-Wan never did “own” R2 or C-3PO. Obi-Wan then makes a comment that suggests he's more than aware of R2's impressive capabilities.
Darth Vader acts like he doesn't even know C-3PO or R2D2. They were with him since he was a child. He did everything he could to forget his past. Also, his plating had been replaced many times, and protocol droids all look very similar.
Because as far as he knows it's just a generic protocol droid. There are millions of them all over the galaxy. He has no reason to think this droid is the same one he was briefly acquainted with many years ago.
C-3PO's memories were (against his own wishes) wiped again on Kijimi when the droidsmith Babu Frik bypassed C-3PO's programming restrictions to allow the droid to translate the Sith writings on a Sith dagger. C-3PO's photoreceptors turned red while he translated the forbidden inscriptions.
Maybe R2-D2 didn't know Darth Vader was Luke's father because he didn't see Palpatine christening him Vader, or maybe he believed Anakin was dead because even Obi-Wan didn't know Anakin was alive.
secondly, in Legends, R2 was given programming that actually restricted him. from sharing any information with Luke or Leia. about their parents or relation to one another. this was all part of the plan. to conceal this information from the twins.
Darth Vader's true identity as Anakin Skywalker was a highly guarded secret, known only to a select few individuals including Palpatine, Obi-Wan Kenobi, Luke, Han, Leia, R2-D2, C-3PO, Master Yoda, Bail Organa, Ahsoka Tano, Darth Maul, Grand Moff Tarkin, Grand Admiral Thrawn, Jocasta Nu, Ferren Barr, Verla, Reva, ...
Since Bail Organa had C-3PO's memory wiped previously in order to prevent him from spilling the secret of Leia's parentage, it could be that this was a semi-regular occurrence for the droid given he couldn't be trusted to keep quiet about things.
The question being answered is: “Do R2D2 and C3PO know that Darth Vader is Anakin?” C-3PO? Definitely not. As many other have pointed out, C-3PO had his memory wiped at the end of Episode III, so it is almost certain that he does not know.
She senses Anakin when she reaches out to the pilot — and Vader realizes that Ahsoka, his former Padawan, lives on. In Season 2, Episode 18, “Shroud of Darkness,” Ahsoka's guilt manifests as a vision of Anakin, who's angry with her for leaving him all those years ago.
Unfortunately, Rex and Vader never meet again though. Why didn't Palpatine feel afraid of showing his evil actions, behaviors and ways to Anakin and the Jedi anymore?
R2 was later able to restore his companion during the Battle of Geonosis. After the battle, R2-D2 and C-3PO were the sole witnesses to the marriage of Anakin Skywalker and Padmé Amidala in Lake Country. After this Padmé gave ownership of R2 to Anakin while she kept 3PO.
In Star Wars: The Force Awakens, set approximately 30 years after Return of the Jedi, R2-D2 is revealed to be kept in storage at the Resistance base on the planet D'Qar, having put himself in a low-power mode after Luke Skywalker's disappearance.
In the years before the Clone Wars, R2-D2 served the Royal House of Naboo. He was aboard Queen Amidala's royal starship when she tried to run the Trade Federation's blockade. Artoo repaired the ship's damaged shields under heavy fire, allowing it to escape into hyperspace.
Three months after the Battle of Endor, Skywalker used the lightsaber during a mission to an Imperial research base on the planet Vetine in order to retrieve the last remaining fragment of a Force-sensitive tree. Skywalker gave his lightsaber to R2-D2 for hiding as they entered the Imperial base.
After decades of missions with R2-D2, Luke Skywalker was reasonably fluent in droidspeak. The most common way for binary droids to communicate with organics was by writing, conveyed via another computer system. This is how astromech droid copilots would speak with the pilot of a starfighter.
